Schur-weyl duality for deligne categories II: The limit case

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2 Scopus citations

Abstract

This paper is a continuation of a previous paper by the author (Int. Math. Res. Not. 2015:18 (2015), 8959-9060), which gave an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (a vector space V with a chosen nonzero vector 1), we constructed in that paper a complex tensor power of V: an Ind-object of the Deligne category Rep(Sν) which is a Harish-Chandra module for the pair (gl(V), P1), where P1 ⊂ GL(V) is the mirabolic subgroup preserving the vector 1. This construction allowed us to obtain an exact contravariant functor SŴν, V from the category Repab(Sν) (the abelian envelope of the category Rep(Sν) to a certain localization of the parabolic category O associated with the pair (gl(V), P1). In this paper, we consider the case when V=C. We define the appropriate version of the parabolic category O and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories Ôp v,C N with N tending to infinity. The Schur-Weyl functors SWv,C N then give an antiequivalence between this category and the category Repab(Sν). This duality provides an unexpected tensor structure on the category Ôp∞ v,C∞.

Original languageEnglish
Pages (from-to)185-224
Number of pages40
JournalPacific Journal of Mathematics
Volume285
Issue number1
DOIs
StatePublished - 1 Jan 2016
Externally publishedYes

Keywords

  • Deligne categories
  • Limits of categories
  • Parabolic category O
  • Schur-Weyl duality

ASJC Scopus subject areas

  • General Mathematics

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